I have the following problem:
Let $I, J ⊂ S $ be ideals and $<$ a monomial order on $S$. Let $G, G'$ be Gröbner bases of $I$, respectively $J$, with respect to $<$ . Prove that if $\,\operatorname{in}_<(g)$ and $\,\operatorname{in}_<(g')$ are relatively prime for any $g ∈ G$, $g' ∈ G'$, then $G ∪ G'$ is a Gröbner basis of $I + J$.
I would be grateful if someone would help me with some advice.